Solve the equations given above by completing the square of perfect square method.
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Answers
Step-by-step explanation:
1. in picture
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2. Given quadratic equation is
x² + x - 20 = 0
x² + x = 20
x² + 2 × x × 1/2 = 20
x² + 2 × x × 1/2 + ( 1/2 )² = 20 + ( 1/2 )²
( x + 1/2 )² = 20 + 1/4
( x + 1/2 )² = ( 80 + 1 )/4
( x + 1/2 )² = 81/4
x + 1/2 = ± √( 81/4 )
x + 1/2 = ± 9/2
x = -1/2 ± 9/2
x = ( -1/2 + 9/2 ) or x = ( -1/2 - 9/2 )
x = 8/2 or x = -10/2
x = 4 or x = -5
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3.The form for a perfect square
is a2+2ab+b2=(a+b)2.
x2+2x−5=0.
Add 5 to both sides of the equation.
x2+2x=5
Divide the coefficient of the x value by 2, then
square the result.
22=1; 12=1
Add the result to both sides.
x2+2x+1=5+1 =
x2+2x+1=6
The left side is now a perfect square trinomial.
x2+2x+1=(x+1)2
(x+1)2=6
Take the square root on both sides.
x+1=±√6
Subtract 1 from both sides.
x=√6−1
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4. We have,
=> m² - 5m = -3
=> m² - 5m + (5/2)² = -3 + 25/4
=> (m - 5/2)² = 13/4
=> m - 5/2 = +-√13/4
=> m - 5/2 = +-√13/2
Case 1
=> m = (√13+5)/2
Case 2
=> m = (5 - √13)/2
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5. 9y2-12y+2=0
Divide equation by 9:
y2-4/3y+2/9=0
y2-4/3y=-2/9
y2-4/3y+square of (2/3)=-2/9+ square of(2/3).
After 1 step solving,
square of (y-2/3)=2/9
y-2/3=+-root 2/9
y=2/3+-root 2/9
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6. QUADRATIC EQUATIONS ⭐
2y² + 9y + 10 = 0
=> 2 ( y² + 9/2y + 5) = 0
=> y² + 9/2y + 5 = 0
=> y² + 9/2y = -5
=> y² + 9/2y + 81/¹6 = -5 + 81/4
=> (y + 9/4)² = 1/16
=> (y + 9/4) = ± 1/4
=> (y + 9/4) = 1/4 or (y + 9/4) = -1/4
=> y = -8/4 or y = -10/4
=> y = -2 or y = -5/2
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7.Given Quadratic equation is
5x² = 4x + 7
=> 5x² - 4x = 7
Divide each term with 5 , we get
x² - ( 4/5 ) x = 7/5
=> x² - 2 × x × ( 2/5 ) = 7/5
=> x²-2(x)(2/5)+(2/5)² = 7/5+(2/5)²
=> ( x - 2/5 )² = 7/5 + 4/25
=> ( x - 2/5 )² = ( 35 + 4 )/25
=> ( x - 2/5 )² = 39/25
=> x - 2/5 = ± √(39/25)
=> x = 2/5 ± √39/5
=> x = ( 2 ± √39 )/5
Therefore ,
x = ( 2 + √39)/5
Or
x = ( 2 - √39 )/5
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Hopefully understandable ^-^.
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